Geometry of asymptotic bias reduction of plug-in estimators with adjusted likelihood

A geometric framework to improve a plug-in estimator in terms of asymptotic bias is developed. It is based on an adjustment of a likelihood, that is, multiplying a non-random function of the parameter, called the adjustment factor, to the likelihood. The condition for the second-order asymptotic unbiasedness (no bias up to $O(n^{-1})$ for a sample of size $n$) is derived. Bias of a plug-in estimator emerges as departure from a kind of harmonicity of the function of the plug-in estimator, and the adjustment of the likelihood is equivalent to modify the model manifold such that the departure from the harmonicity is canceled out. The adjustment is achieved by solving a partial differential equation. In some cases the adjustment factor is given as an explicit integral. Especially, if a plug-in estimator is a function of the geodesic distance, an explicit representation in terms of the geodesic distance is available, thanks to differential geometric techniques for solving partial differential equations. As an example of the adjustment factor, the Jeffreys prior is specifically discussed. Some illustrative examples are provided.